We are now ready to prove Theorem REF . We learn this argument from Orponen in [1]}.
[Proof of Theorem REF ]
Using Frostman's lemma ([2]}), we can find probability measure \(\mu _A\)  supported on \(A\)  such that
\(\mu _A(B(x, r))\lesssim r^\alpha , \forall x\in {\mathbb {R}}, r>0.\) 
